Can I invert only the vertex normals?
I finally solved the issue. The solution is to split the normals, which I assume means telling Blender to stop the 'connection' between the two types of normals, face and vertex:


Enable 'Custom Normal Tools' in Add-Ons
Create an inverted sphere
Enable 'Auto Smooth' in Object Data
Add the 'Normal Edit' modifier to the inverted sphere
Add an empty object in the center of the sphere
Use Empty object as target in the Normal Edit Modifier with the following settings:


Radial
Mix Mode: Subtract
Mix Factor: 1.000



Source: Custom Split Normals (Blender wiki)